Calculation of Expected Time (te):
In PERT (Program Evaluation and Review Technique), the expected time (te) is calculated using the formula:
te = (to + 4tm + tp) / 6

Given:
to = 9 days
tm = 15 days
tp = 21 days

Substituting the values in the formula:
te = (9 + 4(15) + 21) / 6
te = (9 + 60 + 21) / 6
te = 90 / 6
te = 15 days

Calculation of Variance (σ^2):
The variance (σ^2) is calculated using the formula:
σ^2 = ((tp - to) / 6)^2

Given:
to = 9 days
tp = 21 days

Substituting the values in the formula:
σ^2 = ((21 - 9) / 6)^2
σ^2 = (12 / 6)^2
σ^2 = 2^2
σ^2 = 4

Calculation of Standard Deviation (σ):
The standard deviation (σ) is the square root of the variance (σ^2).

Taking the square root of σ^2:
σ = √4
σ = 2

Calculating the Probability:
To calculate the probability of completing the activity within a specific time, we can use the normal distribution.

Given:
te = 15 days
σ = 2 days

We need to calculate the probability of completing the activity in 14 days.

Using the Z-score formula:
Z = (X - te) / σ

Substituting the values:
Z = (14 - 15) / 2
Z = -1 / 2
Z = -0.5

Using the Z-table:
We can refer to a Z-table to find the corresponding probability.

The Z-table gives us the probability for a given Z-score. In this case, we need to find the probability for a Z-score of -0.5.

The Z-table provides the area under the normal distribution curve to the left of a given Z-score. To find the probability of completing the activity in 14 days, we need to find the area to the left of -0.5 on the Z-table.

The Z-table shows that the area to the left of -0.5 is approximately 0.3085.

To find the probability of completing the activity in 14 days, we subtract the area to the left of -0.5 from 0.5 (which represents 50% of the area under the curve).

Probability = 0.5 - 0.3085
Probability = 0.1915

Converting the probability to a percentage:
Probability = 0.1915 * 100
Probability = 19.15%

Therefore, the approximate probability of completing this activity in 14 days is 19.15%, which can be rounded to 19% or approximately 20%.